text: The S-matrix is defined to be the inner product
  of the ath and bth Heisenberg picture asymptotic states.  One may obtain a formula relating the S-matrix to the potential V using the above contour integral strategy, but this time switching the roles of  and .  As a result, the contour now does pick up the energy pole.  This can be related to the s if one uses the S-matrix to swap the two s.  Identifying the coefficients of the s on both sides of the equation one finds the desired formula relating S' to the potential
  In the Born approximation, corresponding to first order perturbation theory, one replaces this last  with the corresponding eigenfunction  of the free Hamiltonian , yielding
  which expresses the S-matrix entirely in terms of V and free Hamiltonian eigenfunctions.
  These formulas may in turn be used to calculate the reaction rate of the process , which is equal to